Hilbert series of subspace arrangements

Derksen, Harm

doi:10.1016/j.jpaa.2006.05.032

Cited by 34 publications

(48 citation statements)

References 18 publications

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“…= dim(J i ). Then one can show that when the arrangement is transversal, one has h I (i) = h J (i) for all i ≥ n. A more complete development is given in [13].…”

Section: Definition 211 (Hilbert Function) the Hilbert Function Of mentioning

confidence: 99%

“…A recursive formula for the Hilbert series of J(A) was given in [13]. Surprisingly, this formula depends only on the codimensions of the intersections (c S , S ⊆ {1, 2, .…”

Section: Definition 211 (Hilbert Function) the Hilbert Function Of mentioning

confidence: 99%

“…13 We choose here real numbers as the basic "units" for measuring complexity in a similar fashion to binary numbers, "bits," traditionally used in algorithmic complexity or coding theory.…”

Section: = Minmentioning

confidence: 99%

“…The importance of hybrid linear models is manifold. First, they are the natural generalizations to single linear models; second, they are sufficiently expressive for representing or approximating arbitrary complex data structures; and third, the understanding of hybrid linear models has been significantly advanced in recent years and many efficient solutions have been developed (see [61,13] and the references therein). Thus, the goal of this paper is to give a comprehensive introduction to some of these new developments in the study of modeling such mixed data with hybrid linear models, and to place many sporadic results in the literature in a coherent and complete mathematical and computational framework.…”

mentioning

confidence: 99%

“…12: end for 13: Segment the remaining samples that are not in the stacks of the highest votes based on the estimated bases. respectively.…”

mentioning

confidence: 99%

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Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data

Ma¹,

Yang²,

Derksen³

et al. 2008

SIAM Rev.

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216

189

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Abstract. Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very complicated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model. In this paper we propose to use subspace arrangements-a union of multiple subspaces-for modeling mixed data: each subspace in the arrangement is used to model just a homogeneous subset of the data. Thus, multiple subspaces together can capture the heterogeneous structures within the data set. In this paper, we give a comprehensive introduction to a new approach for the estimation of subspace arrangements. This is known as generalized principal component analysis (GPCA).In particular, we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust, even when the given data are corrupted by noise or contaminated with outliers. This new method in many ways improves and generalizes extant methods for modeling or clustering mixed data. There have been successful applications of this new method to many real-world problems in computer vision, image processing, and system identification. In this paper, we will examine several of those representative applications. This paper is intended to be expository in nature. However, in order that this may serve as a more complete reference for both theoreticians and practitioners, we take the liberty of filling in several gaps between the theory and the practice in the existing literature.

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“…= dim(J i ). Then one can show that when the arrangement is transversal, one has h I (i) = h J (i) for all i ≥ n. A more complete development is given in [13].…”

Section: Definition 211 (Hilbert Function) the Hilbert Function Of mentioning

confidence: 99%

“…A recursive formula for the Hilbert series of J(A) was given in [13]. Surprisingly, this formula depends only on the codimensions of the intersections (c S , S ⊆ {1, 2, .…”

Section: Definition 211 (Hilbert Function) the Hilbert Function Of mentioning

confidence: 99%

“…13 We choose here real numbers as the basic "units" for measuring complexity in a similar fashion to binary numbers, "bits," traditionally used in algorithmic complexity or coding theory.…”

Section: = Minmentioning

confidence: 99%

mentioning

confidence: 99%

“…12: end for 13: Segment the remaining samples that are not in the stacks of the highest votes based on the estimated bases. respectively.…”

mentioning

confidence: 99%

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